The use of quartz resonators for sensing temperature is known and has generally involved the use of unrotated X- and Y-cut plates. In such devices, the output frequency (f) is a function of temperature (T) which can be expressed by the power series: EQU .DELTA.f/f.sub.0 =(f-f.sub.0)/f.sub.0 =a(.DELTA.T)+b(.DELTA.T).sup.2 +c(.DELTA.T).sup.3, (1)
where .DELTA.T=T-T.sub.0, and f.sub.0 is the reference frequency at a reference temperature T.sub.0 which is normally taken as 25.degree. C. The coefficients a, b and c are referred to as the temperature coefficients of frequency (TCF) of the first-, second- and third-order, respectively, and which are written as TCF.sup.(1), TCF.sup.(2) and TCF.sup.(3). Accordingly: a=TCF.sup.(1), b=TCF.sup.(2), c=TCF.sup.(3). The coefficient "a" comprises the linear term, whereas the coefficient "b" comprises the quadratic or parabolic term, and the coefficient "c" comprises the cubic or cubic parabolic term. Moreover, the units of these coefficients are normally expressed in 10.sup.-6 per kelvin(K), 10.sup.-9 /K.sup.2 and 10.sup.-12 /K.sup.3, respectively.
For unrotated X- and Y-cuts, not only is the TCF.sup.(1) coefficient relatively large, but the coefficients TCF.sup.(2) and TCF.sup.(3) are relatively large as well. While this is indicative of a resonator whose output frequency will be a function of its temperature, the relation between f and T will not be linear and as a result a special calibration is required when it is utilized. Moreover, the calibration is a function of time because of such factors as stress relaxation in the resonator electrodes and changes in the stresses occurring in the crystal mounting supports. These particular perturbations and others go under the collective name of "resonator aging". Additionally, temperature transients supplied to the resonator enclosure also lead to thermal gradients within the crystal and, as a result, produce large frequency excursions which are not predicted by the aforementioned power series since the formula defined by equation (1) governs quasi-static temperature changes.
In view of the inherent limitations existing in non-rotated X- and Y-cut quartz plates, a doubly rotated LC (linear coefficient) cut has been developed. As is well known, the LC cut resonator comprises a cut which is located at an orientation where second-order and third-order terms TCF.sup.(2), TCF.sup.(3) are equal and substantially equal to zero or at least negligible, while the first-order term TCF.sup.(1) is not zero. Such a cut exhibits a substantially linear frequency vs. temperature characteristic which can be expressed as: EQU .DELTA.f/f.sub.0 =(f-f.sub.0)/f.sub.0 =a(T-T.sub.0)=a(.DELTA.T). (2)
The LC cut has not only been used but has been found to be commercially successful in thermometric sensing applications. One such application comprises a quartz thermometer apparatus.
While this type of resonator constituted a distinct advancement in the art, it nevertheless still suffers from certain other inherent limitations such as: the crystal is difficult to manufacture because the two orientation angles .phi. and .theta., polar angles of the spherical coordinate system (0-s, .theta..sub.0, and .phi..sub.0), are such that the cut is not located near any X-ray planes of any reasonable strength, it is uncompensated for any in-plane stresses such as electrode stresses which lead to component aging; it is not compensated for thermal transients which lead to inaccurate frequency and hence temperature readout in the presence of thermal gradients; and, is not compensated against thermal hysteresis which leads to temperature ambiguities which are dependent upon the size and direction of the temperature excursions sensed by the resonator.